Extended finite element and level set methods with applications to growth of cracks and biofilms

The eXtended Finite Element Method (X-FEM) coupled with the level set method is used to assess the damage tolerance of complex 3D components and the growth of two-dimensional biofilms. The Finite Element Method (FEM) has been used for decades to solve a myriad of problems. However, the FEM is not adequate to solve problems with features such as interior boundaries, discontinuities or singularities, because of the need of remeshing and high mesh densities.; In X-FEM, the standard FE approximation space is enriched with specially tailored functions to help capture the challenging features of a problem. Enrichment functions may be discontinuous (to model discontinuities in the field), their derivatives can be discontinuous (to model kinks in the field), or they can be chosen to incorporate a known characteristic of the solution (such as the square root singularity of linear elastic fracture mechanics).; The level set (LS) method allows for the treatment of internal boundaries and interfaces by the sole construction of an implicit function, thereby tracking interfaces, their merging, cusps, intersection with boundaries without any explicit treatment of the interface geometry.; This hybrid X-FEM/LS Method is used to model crack growth in complex three-dimensional components and to assess their damage tolerance. A superelement/X-FEM/level set method where a C++ implementation of the X-FEM is interfaced with a commercial code to allow the treatment of fatigue crack growth in complex structures is proposed. It is found that the method permits an accurate determination of stress intensity factors and the assessment of the damage tolerance of three-dimensional components that may be modeled with several element types.; A hybrid X-FE/LSM is developed to treat the growth of biofilms in two dimensions. X-FEM is used to account for the presence of the biofilm through discontinuous and asymptotic enrichment. The geometry of the film is numerically described using a level set method, which tracks the discontinuity surface by solving a hyperbolic conservation law. The biofilm can be grown through a fixed mesh, which suppresses the need for remeshing the discontinuity at each step. Numerical examples are presented to illustrate the method.